dolls - slot method

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vikram4689: Legendary Member; Posts: 1325; Joined: Sun Nov 01, 2009 6:24 amFollowed by: 14 members; Thanked: 105 times

by vikram4689 » Tue Aug 07, 2012 8:49 am

Gordon buys 5 dolls for his 5 nieces. The gifts include two identical S beach dolls, one E, one G, one T doll. If the youngest niece doesn't want the G doll, in how many different ways can he give the gifts?

is my application of slot method correct ?

lets treat 2 identical dolls as different
since youngest girl has restrictions, fulfill her choice first.
youngest girl can get the dolls in 4 ways
then all other girls can be given dolls in 4*3*2*1 ways = 24 ways
total no. of ways = 4*24=96

Now, since there are 2 identical dolls, we need to divide(reduce) above arrangements by no. of ways identical dolls affected(increased) above arrangements i.e. 2!=2

Hence, total arrangements = 96/2=48

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Source: Beat The GMAT — Problem Solving | Tue Aug 07, 2012 8:49 am

Kobe_Kassidy: Junior | Next Rank: 30 Posts; Posts: 15; Joined: Mon Aug 06, 2012 8:06 am; Thanked: 1 times

by Kobe_Kassidy » Tue Aug 07, 2012 9:25 am

I see how you used your method. Looks fine to me.

Just to add a different perspective, here is what I did.

I assumed no restrictions and distributed the 5 dolls among the 5 kids (5 ways for the first, 4 ways for the second, 3 ways for the third,......1 way for the 5th)

This gives you 5!, which is 120. However, since two dolls are identical, you have to recognize that this method will count the possibilities where the two girls with the beach dolls would essentially swap the same doll, which would not produce a unique arrangement. Therefore, you divide 120 by 2 and get 60

Now, take out the restriction, which means finding the number of arrangements where the youngest niece would end up with the G doll, and then subtracting that number.

G doll would be distributed only 1 way (youngest niece), the remaining dolls can be distributed in 4,3,2, and 1 ways respectively. So 4! x 1 = 24. Again, however, you need to divide by 2 to eliminate the additional arrangements caused by the two beach dolls. 24/2 = 12

60 - 12 = 48 possible combinations

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